Decay Constant & Half-Life (HL) (HL IB Physics)

• Since radioactive decay is spontaneous and random, it is useful to consider the average number of nuclei that are expected to decay per unit time
  • This is known as the average decay rate

• As a result, each radioactive element can be assigned a decay constant
• The decay constant λ is defined as:

 The probability that an individual nucleus will decay per unit of time

• When a sample is highly radioactive, this means the number of decays per unit time is very high
  • This suggests it has a high level of activity

• Activity, or the number of decays per unit time can be calculated using:

• Where:
  • A = activity of the sample (Bq)
  • ΔN = number of decayed nuclei
  • Δt = time interval (s)
  • λ = decay constant (s^-1)
  • N = number of nuclei remaining in a sample

• The activity of a sample is measured in Becquerels (Bq)
  • An activity of 1 Bq is equal to one decay per second, or 1 s^-1

• This equation shows:
  • The greater the decay constant, the greater the activity of the sample
  • The activity depends on the number of undecayed nuclei remaining in the sample
  • The minus sign indicates that the number of nuclei remaining decreases with time

• Half-life and decay constant can be linked, using an equation called the exponential decay equation showing how the number of active nuclei, N, changes over time, t, where N₀ is the initial number of active nuclei

N = N₀ e^–λt

• When time t is equal to the half-life t_½, the number of active nuclei in the sample, N, will be half of its original value, so N = ½ N₀

• The formula linking half-life and decay constant can then be derived as follows:

• Therefore, half-life t_½ can be calculated using the equation:

• This equation shows that half-life t_½ and the radioactive decay rate constant λ are inversely proportional
  • Therefore, the shorter the half-life, the larger the decay constant and the faster the decay

• The half-life of a radioactive substance can be determined from decay curves and log graphs
• Since half-life is the time taken for the initial number of nuclei, or activity, to reduce by half, it can be found by
  • Drawing a line to the curve at the point where the activity has dropped to half of its original value
  • Drawing a line from the curve to the time axis, this is the half-life

A linear decay curve. This represents the relationship:

• Straight-line graphs tend to be more useful than curves for interpreting data
  • Due to the exponential nature of radioactive decay logarithms can be used to achieve a straight line graph

• Take the exponential decay equation for the number of nuclei

N = N₀ e^–λt

• Taking the natural logs of both sides

ln N = ln (N₀) − λt

ln N = − λt + ln (N₀) 

• In this form, this equation can be compared to the equation of a straight line

y = mx + c

• Where:
  • ln (N) is plotted on the y-axis
  • t is plotted on the x-axis
  • gradient = −λ
  • y-intercept = ln (N₀)

• Half-lives can be found in a similar way to the decay curve but the intervals will be regular as shown below:

A logarithmic graph. This represents the relationship:

• Note: experimentally, the measurement generally taken is the count rate of the source
  • Since count rate ∝ activity ∝ number of nuclei, the graphs will all take the same shapes when plotted against time (or number of half-lives) linearly or logarithmically